NOTES ON
REAL VALUED FUNCTIONS OF A REAL VARIABLE
Sanjeev Agrawal
04: SERIES 2
Tests for non-negative series
More often than not, given an infinite series, it is very difficult to find the right comparison series
to determine its convergence behaviour.
Hence tests are developed that do not use any external sequences.
We first take a look at the two most important tests.
1
A: Cauchy’s Root Test
A01 Version 1:
Hypothesis:
(1)
R.
n
(2) lim
|
n
n|
1
n
l
Conclusion:
l
1
l
1
n
converges absolutely.
0&
n
a n diverges.
1, then this test fails to give any information about
If l
n.
.............................
A02 Version 2:
Hypothesis:
(1)
(2)
(3)
R.
n
K:
n
0, n
lim sup a n
1
n
K.
l
Conclusion:
l
1
l
1
If l
n
n
converges;
0&
a n diverges.
1, then this test fails to give any information about
n.
.............................
A03 Note:
A03.01: For submissions: This test can be used without providing a proof, but only after
stating the version completely & clearly and then applying it properly. The proof of this test,
either version, may be asked in the exams.
A03.02: Version 1 follows from Version 2 (How?). I will prove Version 2 in class. Modify that
proof to provide a proof of Version 1 without using lim sup.
.............................
2
B: D’Alembert’s Ratio Test
B01 Version 1:
Hypothesis:
(1)
R.
n
(2)
K:
(3)
0, n
n
n
lim
n
K.
.
n 1
Conclusion:
1
n
1
If
n
converges absolutely.
0&
n
diverges.
1, then this test fails to give any information about
n.
.............................
B02 Version 2:
Hypothesis:
(1)
(2)
n
K:
n
R.
0, n
K.
Conclusion:
lim inf aann1
lim sup aann1
1
1
an
0&
a n diverges;
a n converges;
1 lim sup aann1
The test fails if lim inf aann1
.............................
B03 Note:
(i) For submissions: This test can be used without providing a proof, but only
after stating the version completely & clearly and then applying it properly. The proof of this
test, either version, may be asked in the exams.
(ii) Version 1 follows from Version 2 (How?). I will prove Version 2 in class.
Modify that proof to provide a proof of Version 1 without using lim sup.
(iii) For submissions: You may find some other versions of these, and other
tests. If you use any version (whether stated here or not) whose proof is not given in
claproofsss, you must provide a proof. It may be advisable to prepare an Appendix with these
proofs so that you can use them multiple times in Assignments and Quizzes. Proper application
is a must.Do not use any version in the exams without fully stating it. The exact version mpart
of the submitted Appendix.
.............................
3
B04 Note:
B04.01: Whenever the tests tell you that a series is divergent, it is because the conditions in that
case imply that a n
0.
This becomes absolutely crucial in most applications of the test.
B04.02: Since most series are not (ultimately) non-negative, very often the test is applied to
|a n |.
Thus, the tests either tell us that a n is either abosuletly convergent or not.
B04.03: If the test tells that |a n | diverges, a n may still be conditionally convergent.
0 and hence
But divergence criterion tell us that |a n | 0. But then a n
a n cannot even be conditionally convergent.
B04.04: In both cases, “test fails” means that there examples of both convergent and divegent
series satisfying the given conditions.
B04.05: Cauchy’s Second Theorem on Limits ( See the notes on limit superior and limit inferior
of sequences) shows that the Root Test is more powerful than the Ratio Test, in the sense that
whenever the Ratio Test works, so will the Root Test; but sometimes the Root Test works whereas
the Ratio Test fails. For example, look at
1
1
1
1
1
1 ...
2
2
3
2
3
2
3
2
33
...............................................................................
4
C Raabe’s Ratio Test:
C01 Raabe’s Ratio Test - Version 1::
Hypothesis:
(1)
(2)
R.
xn
K : xn
0, n
K.
1
k.
(3) lim
n xxnn1
n
Conclusion:
k
1
x n converges (absolutely).
k
1
x n diverges.
If k
1, then this test fails to give any information about
xn.
.............................
C02 Raabe’s Ratio Test - Version 2:
Hypothesis:
(1)
yn
(2)
(3)
K : yn
1&M
N:
R.
0, n K.
yn 1
1 n. n
yn
M.
Conclusion:
y n converges absolutely.
.............................
C03 Raabe’s Ratio Test - Version 3:
Hypothesis:
(1)
yn
(2)
(3)
K : yn
1&M
N:
R.
0, n K.
yn 1
1 n. n
yn
Conclusion:
y n does not converge absolutely.
Note: y n may converge conditionally.
.............................
5
M.
C04 Raabe’s Ratio Test - Version 4:
Hypothesis:
(1)
R.
zn
K : zn
(2)
(3) lim
n
n
0, n
zn
zn 1
1
K.
p.
Conclusion:
p
1
0
p
p
0
If p
z n converges absolutely.
1
either
z n diverges or it converges conditionally.
z n diverges.
1, then this test fails to give any information about
zn.
Note: The case 0 p 1 has to be settled.
.............................
C05 Lemma on null sequences:
Hypothesis:
(1)
R.
wn
K : w n 0, n
(3) lim
n wwnn1
1
n
(2)
K.
q.
Conclusion:
q
0
wn
0.
bn
R.
.............................
C06 Raabe’s Ratio Test - Version 5:
Hypothesis:
(1)
(2)
0, n.
bn
(3) lim
n
n
bn
bn 1
1
n 1
bn
b1
1
n 1
b n converges conditionally.
1
p.
Conclusion:
0
p
1
b2
. . . converges.
n 1
0
p
1
n 1
.............................
6
C06 Raabe’s Ratio Test - Version 6:
Hypothesis:
(1)
an
R.
0, n.
(2) a n
Conclusion:
. lim inf n aann1
lim sup n aann1
1
1
a n converges.
1
1
a n diverges.
.............................
C07 Note:
C07.01: Version 1 follows from Version 6.
C07.02: Versions 2 & 3 (combined) are stronger than Versions 6 & 1, but the latter are easier to
apply.
C07.03: Versions 4 & 5 give partial answers in case the others do not work.
C07.04: Version 6. and if time permits, Versions 2 & 3 will be proved in class.
C07.05: For submissions: Version 1 (after providing a proof) can be used. Versions 4 and 5
can be used without proof, but with a lower weightage if simpler tests can provide the same
conclusion. Version 1 may be asked in the exams.
C07.06: For submissions: Versions 2 and 3 can be used only if you can explain the proof.
.............................
7
D Logarthim Test
D01 Logarthim Test - Version 1::
Hypothesis:
(1)
(2)
(3)
R.
xn
K : xn
0, n
lim
n log xxnn1
n
K.
k.
Conclusion:
k
1
x n converges (absolutely).
k
1
x n diverges.
If k
1, then this test fails to give any information about
xn.
.............................
D02 Logarthim Test - Version 2::
Hypothesis:
(1)
(2)
an
K : an
R.
0, n
K.
Conclusion:
If lim inf n log aann1
k 1, then
l 1, then
If lim sup n log aann1
In all other cases, the test fails.
a n converges.
a n diverges.
.............................
D03 Note:
D03.01: Version 1 follows from Version 2.
D03.02: Many state this for ‘ a n be an ultimately postive decreasing real sequence’. This is so
that log aann1 is ultimately positive.
We dont, because if some of log aann1 are negative, and
(i) if the limit is positive, then ultimately log aann1 is positive.
(ii) if the limit is negative, then ultimately log aann1 is negative. That implies a n is an
ultimately postive increasing real sequence & hence a n diverges.
D03.03: The proof is not as simple as that of the other two tests.
D03.04: The test is useful if lim aann1 nvolves e.
D03.05: If time permits, Version 2 will be proved in class.
8
D03.06: For submissions: This test can be used without providing a proof, but only after
stating it clearly and applying it properly. Only elementary properties of the logarthim
function can be used. Power series and other expansions are not allowed. The proof of this test
will not be asked in the exams.
.............................
9
E Cauchy’s Condensation Test
E01 Cauchy’s Condensation Test::
Hypothesis:
(1)
(2)
0
an
an 1, n
(3) a n
(iv)
R.
an
0.
an
Conclusion:
2 k a 2 k behave in a similar manner, i.e., either both converge or both diverge.
a n and
n 1
k 1
.............................
E02 Note:
E02.01: Try this test on series with a n 1
n log n
p
or n log n log log n
p
or ....
E02.02: The test will be proved in class.
E02.03: For submissions: This test can be used without providing a proof, but only after
stating it clearly and applying it properly. The proof of this test may be asked in the exams.
.............................
10
F Integral Test
F01 Integral Test::
Let f x be a positive decreasing (real) function for x
1 that vanishes at .
Let
n
fk
Sn
k 1
and
n
In
f
1
and
Dn
In.
Sn
Then
0
fn
1
Dn
lim
D n exists (
n
1
Dn
f 1 ; for n
1, 2, 3, . . .
).
f n converges iff lim
I n exists.
n
n 1
0
Dk
lim
Dn
n
f k ; for k
1, 2, 3, . . .
.
k 1
fk
1
f
f k ; for k
1, 2, 3, . . .
k
f : lim
In
n
1
...............................................................................
F02 Note:
F02.01: For submissions: This test can be used without providing a proof, but only after
stating it clearly and applying it properly. The proof of this test may be asked in the exams.
.............................
11
There are several other ratio tests and other tests that are available for infinite series. We mention
some here.
G01 Kummer’s Test: Let a n and b n be ultimately postive real sequences such that
diverges.
(i) k
1 an
1
k. Then
bn an 1
bn 1
0
a n converges;
(ii) k
0
bn
Let lim
a n diverges.
G02 Gauss’s Test: Let a n be an ultimately postive real sequence such that
an
an 1
1
n
O
1
n
1
, where
0.
Then
(i)
1
a n converges;
(ii)
1
a n diverges.
Gauss’s Test, if you can find the
and , will work wherever most other ratio tests work.
For submissions: Kummer and Gauss tests cannot be used. Their proofs will not be asked
in the exams.
Here “O” is the ‘Big OH’.
Remember:
If a n and b n are two sequences, we say that a n
M, N such that
|a n | M|b n |, n
...............................................................................
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O bn
N.
iff there exist positive real numbers